1. Can the Hurst Exponent be used to detect Levy Flights?
Nate Bradley
CVEN-6833
Dec. 16, 2005

2. Sediment Dispersion
Can we describe the movement of fluvial sediment statistically?
Dispersion of landslide debris by streams
Transport of solid-phase contaminants
Modeling the cosmogenic radionuclide inheritance of water-borne sediment

3. Modes of Sediment Dispersion
Normal Dispersion
Described by the ADE
Well defined mean and variance
Peak concentration and center of mass correspond
Anomalous Dispersion
Described by a fractional ADE
Large variance of spatial distribution
Center of mass well ahead of peak concentration.

4. Sediment Dispersion as a Random Walk
Particles move a random distance at each iteration.
Distance traveled per iteration, “hop length,” chosen from a PDF.
The type of PDF governing particle motion controls the nature of the dispersion
Example: Brownian Motion

5. Random Walks Normal vs. Anomalous Dispersion

6. What does this have to do with sediment?
Particles may rest on flood plains or bars for long periods of time and then move rapidly during rare, large floods.
Heavy-tailed distributions don’t just apply to hop length. Sediment flux could be heavy-tailed.
We need statistical tools besides variance to investigate the nature of dispersion.

7. The Hurst Exponent in Geophysical Time Series
Developed by H.E Hurst in 1951 while modeling reservoir size.
The Hurst exponent is the slope of the Rescaled Range on a log-log plot
In reservoir design, the Rescaled Range is a measure of the reservoir capacity needed to maintain a constant release flow during wettest and driest periods.
For a purely random process, H ~ 0.5
For many geophysical time series (precipitation, stream flow, sediment flux), H > 0.5. This is the Hurst Phenomenon.
H > 0.5 implies that natural processes are not random. They have a “long memory.”

8. Suspended Sediment Time Series
Colorado River at Lee’s Ferry from Oct. 31, 1947 to Sept. 30, 1965
Mississippi River at Tarbert from Oct. 1, 1949 to Sept. 30, 1975
H > 0.5. Does “long memory” mean that extreme, rare events can dominate a statistic over long time intervals?

9. The Rescaled Range
Choose a series of windows to divide data. For a N=100 element time series, n could be [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
For each window size, n, divide data into N-n+1 overlapping regions. (1-10), (2-11),…, (91,100).
For each region, calculate and save the cumulative sum(Xi - <X>) for that region. This is S*.
The Rescaled Range for that region is max(S*) - min(S*) divided by std. dev. for the region.
Each value of n will have N-n+1 Rescaled Range values. Plot Rescaled Range vs. n on a log-log scale.

10. Hurst Exponent for Random Walks
Normal distribution of hop lengths should result in H~0.5.
Levy Flights, a random walk with a power law distribution of hop lengths might have H > 0.5 because the occasional very large hop dominates. A “long memory process.”

11. Results (You can’t always get what you want.)
The Hurst Exponent cannot detect Levy Flights or behavior governed only by a heavy-tailed distribution.
Rescaled range corrects for Levy flight behavior because of the standard deviation in denominator. Standard deviation scales as >0.5 for Levy Flights.